MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 6 pps

GEGENBAUER FUNCTIONS (TYPE A)
153
9.8
GEGENBAUER
FUNCTIONS
(TYPE
A)
The Gegenbauer equation in general
is
given
as
d2C;' (x)
dC2'
(x)
dx2
dx
-
(2X'
+
1)x-
+
n(n
+
2X')C,"'(x)
=
0.
(9.250)
(1
-
x2)
For

X
=
1/2
this equation reduces to the Legendre equation.
values
of
n
its solutions reduce to the Gegenbauer
or
Legendre polynomials:
For
integer
(9.251)
In the study
of
surface oscillations
of
a
hypersphere one encounters the equa-
tion
-
(2m
+
3)x- dU,"(x)
+
Xurn
A(
>=o,
(9.252)
d2U," (x)

dx2
dx
(1
-x2)
solutions of which could
be
expressed in terms
of
the Gegenbauer polynomials
as
where
X
=
(1
-
m)(l+
m
+
2).
(9.254)
Using
x
=
-cos6
UT(X)
=
Z?(Q)(sinQ)-rn-l
we can put Equation
(9.252)
into the second canonical form as

(9.255)
(9.256)
(9.257)
m(m
+
1)
dQ2
On the introduction
of
A"
=
x
+
(m
+
I)~,
(9.258)
and comparing with, Equation
(9.90),
this is
of
type
A
with
c
=
p
=
d
=

0,
a
=
1,
and
z
=
8,
and its factorization is given by
k(6,
m)
=
m
cot
B
p(m)
=
m2.
(9.259)
(9.260)
154
STURM-LIOUVILLE SYSTEMS AND THE FACTORIZATION METHOD
The solutions are found by using
and the formulas
(9.261)
(9.262)
Note that
Z[n
is the eigenfunction corresponding to the eigenvalue
A"=

(2+112, 1-m=0,1,2
, ,
(9.263)
that is,
to
x
=
(1
+
1)2
-
(m
+
1)2
=
(2
-
m)(l+ m
+
2).
(9.264)
9.9
SYMMETRIC
TOP
(TYPE
A)
The wave equation for
a
symmetric top is encountered in the study
of

simple
molecules.
If
we separate the wave function
as
U
=
@(U)
exp(iK4) exp(im+), (9.265)
where
8,4,
and
+
are the Euler angles and K and m are integers,
O(8)
satisfies
the second-order ordinary differential equation
dO(8)
(m
-
KCOs8)2
@(e)
+
a@(U)
=
0,
(9.266)
&O(U)
de2
+cotu

-
dB
sin2
u
where
(9.267)
A,
W,
C,
and
h
are other constants that come from the physics
of
the problem.
With the substitution
Y
=
O(U)
sin'/2
U,
(9.268)
Equation (9.266) becomes
Y
+
(a
+
K~
+
1/4)Y
=

0.
(9.269)
(m- 1/2)(m+ 1/2)+rc2-2rn~cosU
sin2
u
BESSEL
FUNCTIONS (TYPE C)
155
This equation is of type
A,
and we identify the parameters in Equation
(9.90)
as
a
=
1,
c
=
-1/2,
d
=
-6,
p
=
0.
The factorization
is
now given by
k(0,
m)

=
(m
-
1/2)
cot
e
-
K/
sin
8,
p(m)
=
(m
-
1/2)2.
Eigenfunctions can
be
obtained
from
SinJ-n+l/2
COSJ+n+l/2
-
e
2
2
by using
2
1r2
1
YE-’

=
(J
+
-)2
-
(m
-
-)2
L
The corresponding eigenvalues are
c7
+
K
+
1/4
=
(J
+
1/2)2
J
-
Iml
and
J
-
1.1
=
0,1,2,

so

that
(9.270)
(9.271)
(9.272)
(9.273)
(9.274)
(9.275)
(9.276)
9.10
BESSEL
FUNCTIONS (TYPE
C)
Bessel’s equation is given
as
z2J$(z)
+zJL(z)
+
(A22
-
m2)Jm(.)
=
0.
(9.277)
Multiplying this equation
by
l/z,
we obtain the first canonical form
as
(9.278)
where

p(z)
=
z,
and
~(z)
=
2.
(9.279)
156
STURM-LIOUVILLE SYSTEMS AND THE FACTORIZATION METHOD
A
second transformation,
(9.280)
(9.281)
gives
us
the second canonical form
9
=
0.
1
d21k
(m2
-
1/4)
dx2
22
-+
[x-
(9.282)

This is type
C,
and its factorization is given as
(9.283)
p(m)
=
0.
(9.284)
(m-
f)
k(x,m)
=
~
x'
Because
p(m)
is neither a decreasing nor an increasing function
of
m,
we have
no limit (upper
or
lower)
to
the ladder. We have only the recursion relations
and
where
9,
=
XWm(A1/2X).

9.11
HARMONIC
OSCILLATOR
(TYPE
D)
The Schrodinger equation
for
the harmonic oscillator is given as
(9.285)
(9.286)
(9.287)
(9.288)
where
equation can be written in either
of
the two
forms
(See Problem
9.14)
=
(h/p~)*/~x
and
A
=
2E/hw
in terms
of
the physical variables. This
O-O,@x
=

(A
+
1)Qx
(9.289)
and
0+0_9x
=
(A
-
l)Ikx,
(9.290)
PROBLEMS
157
where
(9.291)
Operating on Equation (9.289) with
Of
and on Equation (9.290) with
0-
we obtain the analog
of
Theorem
I
as
*A,,
0:
o+*x
(9.292)
and
QA-2

a
o-*x.
(9.293)
Moreover, corresponding to Theorem
IV,
we find that we can not lower the
eigenvalue
X
indefinitely. Thus we have
a
bottom of the ladder
A
=
2nf
1,
n
=
0,1,2
,"'
.
(9.294)
Thus the ground
state
must satisfy
o-qo
=
0,
(9.295)
(9.296)
Now the other eigenfunctions can be obtained from

\kn+l
=
[an
+
2]-'/20+*11,,
(9.297)
*,_I
=
[2n]-"20_*,.
(9.298)
Problems
9.1
Starting from the first canonical form
of
the Sturm-Liouville equation:
dx
[P(Z)F]
+
q(2)9(2)
+
XW(Z)*(Z)
=
0,
x
E
[a,
b]
,
derive the second canonical form:
where

d2p1
2
dpdw
1
d2w
pdz dz
w
dz2 pdz2
I
+ +
158
STURM-LIOUVILLE SYSTEMS AND THE FACTORlZATlON METHOD
by using the transformations
Y(Z)
=
w
[W(~)P(~)I~~~
and
W(X)
dz
=
dx
[m]
9.2
Derive the normalization constants in
W+I"
fl(cos6)
and
21
+

1(1
-m)!
1
y,-"(e,$>
=

[L-1"
p,(cos6).
J
2
(i+l)!
27r
9.3
Derive the normalization constant in
9.4
Derive Equation
(9.195),
which
is
given
as
9.5
The general solution of the differential equation
is given
as
the linear combination
y(x)
=
C,
sin

fix
+
C,
cos
Ax.
Show that factorization of this equation leads to the trivial result with
k(x,m)
=
0,
p(m)
=
0,
and the corresponding ladder operators just produce other linear combinations
of sin
Ax
and cos
dz.
9.6
Show that taking
k(z,
m)
=
h(z)
+
kl(z)m
+
k2(z)m2
PROBLEMS
159
does

not lead
to
any new categories, except the trivial solution given in
Prob
lem
9.5.
A
similar argument works for higher powers
of
m.
9.7
in
k(z,
m),
no new factorization types
appear.
9.8
Show that
Show that
as
long
as
we admit
a
finite number
of
negative powers of
m
is
a

periodic function
of
m
with the period one.
Use this result
to
verify
9.9
Derive the stepdown operator in
9.10
the equation
Follow the same procedure used in Path
I
in Section
9.6.5
to
derive
ct
m
1
(1+1-m)
yln,?’(8)
=
-
C,+i,m
(1
+
1
-
m)

J(l
+
m
+
1)(1+
m
+
2)
COS~
7-
(1+;)sin6’}yY(B).
d8
sin8
9.11
tions
of
the first kind:
Use the factorization method
to
show that the spherical Hankel func-
hp
=
j,
+
in,
can be expressed
as
Hint: Introduce
in
yj‘

+
[
1
-
-
”1
y1
=
0.
160
STURM-LIOUVILLE SYSTEMS
AND
THE
FACTORIZATION
METHOD
9.12
normalized eigenfunctions
y(n,
E,
T)
of
the differential equation
Using the factorization method, find
a
recursion relation relating the
to
the eigenfunctions with
1
f
1.

Hint: First show that
1
=
n
-
1,n
-
2,
,
1
=
integer
and the normalization is
9.13
The harmonic oscillator equation
d2
9
dx2
-
+
(E
-
x2)*(x)
=
0
is
a
rather special case
of
the factorization method because the operators

O&
are independent of any parameter.
i) Show that the above equation factorizes
as
d
dx
o,= x
and
d
dx
0-
=

-x.
ii) In particular, show that if
9&(z)
is
a
solution for the energy eigenvalue
E,
then
is
a
solution for
E
+
2,
while
is
a

solution for
E
-
2.
iii) Show that
E
has
a
minimum
with
Emin
=
1,
En
=
2n
+
1,
n
=
0,1,2,
.
. .
PROBLEMS
161
and show that the
E
<
0
eigenvalues

are
not allowed.
to
€,in
and then
use
it to express all the remaining eigenfunctions.
iv) Using the factorization technique, find the eigenfunction corresponding
Hint: Use the identity
9.14
Show that the standard method for the harmonic oscillator problem
leads
to
a
single ladder with each function on the ladder corresponding to
a
different eigenvalue
A.
This follows from the fact that
~(z,
m)
is independent of
m.
The factorization we have introduced in Section
9.11
is
simpler, and in fact
the method of factorization originated from this treatment
of
the problem.

9.15
The spherical Bessel functions
jl(2)
are
related to the solutions
of
d2Yl
l(1
+
1)
-
&2+
[
I
x2
]
Yd2)
=o,
(regular at
x
=
0)
by
Y1
(XI
j&)
=

2
Using the factorization technique, derive recursion formulae

i) Relating
j~(z)
to
j,+,(z)
and
j~-l(z).
ii) Relating
ji(x)
to
j,+,(x)
and
jl-l(z)
.
This Page Intentionally Left Blank
I0
COORDINATES
and
TENSORS
Starting with a coordinate system
is
probably the quickest way to introduce
mathematics into the study of nature. There are many different ways to
choose a coordinate system. Depending on the symmetries of the physical
system, a suitable choice not only simplifies the problem but also makes the
interpretation of the solution easier. Once a coordinate system is chosen, we
can start studying physical processes in terms of mathematical constructs like
scalars, vectors, etc. Naturally the true laws of nature do not depend on the
coordinate system we use; thus we need a way to express them in coordinate
independent formalism. In this regard tensor equations, which preserve their
form under general coordinate transformations, have proven

to
be very useful.
In this chapter we start with the Cartesian coordinates, their transforma-
tions, and Cartesian tensors. We then generalize our discussion to generalized
coordinates and general tensors. The next stop in our discussion
is
the coor-
dinate systems in Minkowski spacetime and their transformation properties.
We also introduce four-tensors in spacetime and discuss covariance of laws
of nature. We finally discuss Maxwell’s equations and their transformation
properties.
10.1
CARTESIAN COORDINATES
In threedimensional Euclidean space a Cartesian coordinate system can be
constructed by choosing three mutually orthogonal straight lines.
A
point is
defined by giving its coordinates,
(q,z2,q),
or by using the position vector
163
164
COORDINATES AND
TENSORS
fig.
10.1
Cartesian coordinate system
+
r
as

f-
r
-
IL&
+
z2G2
+
~3G3
(10.1)
=
(XI
7
22,23),
(10.2)
where
G;
are unit
basis
vectors along the coordinate axis (Fig.
10.1).
Similarly,
an arbitrary vector in Euclidean space can
be
defined
as
+
u
=
U,ZI
+

a2Z2
+
a323,
(10.3)
where the magnitude is given
as
IZi'l
=a
(10.4)
10.1.1
Algebra
of
Vectors
i) Multiplication
of
a
vector with
a
constant
c
is done by multiplying each
component with that constant:
c-2
=
CUlG,
+
ca2Gp
+
cu3S3,
(10.5)

=
(cal,caz,
ca3).
CARTESIAN COORDINATES
165
t
i?=izx*
Fig.
10.2
Scalar
and
vector products
ii) Addition
or
subtraction
is
done by adding
or
subtracting the correspond-
ing components
of
two vectors:
i?*T=
(a1
kbl,azfb2,asfb3).
iii) Multiplication of vectors.
There are two types of vector multiplication:
a)
Dot
or

scalar product
is defined as
(10.6)
(
10.7)
(a,
b)
=
i?.
f
=
abcos@ab
=
aibi
+
a2b2
+
asbs,
where gab is the angle between the two vectors.
b)
Vector product
is defined as
(10.8)
-+
-+.
C=ZX
b
=
(@b3
-

a3b2)sl
+
(~3bl
-
alba)&
+
(albz
-
azbl)Z3
.
Using the permutation symbol, we can
also
write the components of a vector
product
as
3
Ci
=
QjkUjbk,
(10.9)
j,k=l
where the permutation symbol takes the values
+1
for
cyclic permutations
tijk
=
0
when any two indices are equal
.

-
1
for anticyclic permutations
c
=
UbSin6ab
,
(10.10)
{
The vector product of two vectors is again a vector with the magnitude
where the direction is conveniently found by the right-hand rule (Fig.
10.2).
166
COORDINATES AND TENSORS
Fig.
10.3
Motion in Cartesian coordinates
10.1.2
Differentiation
of
Vectors
In
a
Cartesian coordinate system motion of
a
particle can
be
described by
giving its position in terms
of

a
parameter, which is usually taken as the time
(Fig.
10.3),
that
is,
?(t)
=
(Zl(t),
zz(t),
53(t))-
(10.11)
We can now define velocity
3,
and acceleration
3
as
T+(t
+
At)
-
T+(t)
At
-
lim
d7

dt
At-0
7

d3
?(t
+
At)
-
T(t)
At
-
=
lim
d2
?;'
dt
At-0
7
dt2
'
(10.12)
(10.13)
(10.14)
(10.15)
The derivative
of
a
general vector is defined similarly. Generalization of these
equations to
n
dimensions is obvious.
10.2
ORTHOGONAL TRANSFORMATIONS

There are many ways to chose the orientation
of
the Cartesian axes. Symme-
tries
of
the physical system often make certain orientations more advantageous
than others. In general, we need
a
dictionary to translate the coordinates
ORTHOGONAL TRANSFORMATIONS
167
Fig.
10.4
Direction cosines
assigned in one Cartesian system to another.
A
connection between the coor-
dinates
of
the position vector assigned by the two sets
of
Cartesian axes with
a
common origin can be obtained as (Fig.
10.4)
(10.16)
(10.17)
where
This can
also

be written
as
where cos
6ij
are called the direction cosines defined
as
168
COORDINATES AND TENSORS
h
cos
eij
=
isi
.
sj.
(10.20)
Note that the
first
unit basis vector
is
always taken
as
the barred system, that
is,
h
cosej,
=isj
.si.
(10.21)
The transformation equations obtained

for
the position vector are
also
true
for an arbitrary vector
3
as
-
al
=
(cosell)
al
+
(cose12)a2
+
(cose13)a3
a2
=
(cosQ21)
a1
+
(cosQ22)az
+
(cosQ23)a3
a3
=
(cose3,)
al
+
(cos~32)a2

+
(COS~33)a3
.
(10.22)
-
-
The transformation equations given in Equation
(10.22)
are
the special case
of
general linear transformation, which can be written as
-
21
=
allxl
f
a1222
+
a13x3
22
=
a2121
+
a2222
+
a2323
x3
=
a31x1

+
a3222
+
(333x3
.
(10.23)
-
-
aij
are constants independent
of
7
and
7.
A
convenient way
to
write Equa-
tion
(10.23)
is
(10.24)
where we have used the Einstein summation convention, which implies sum-
mation over the repeated (dummy) indices, that is, Equation
(10.24)
means
-
z.
-a,.%.
z

-
t2
3,
i,j=
1,273,
3
=
aijxj,
i,j
=
1,2,3.
(10.25)
(10.26)
Unless otherwise
stated,
we use the Einstein summation convention. Magni-
tude
of
7
in this notation is shown
as
r
=
m.
(10.27)
Using matrices, transformation Equations
(10.23)
can also
be
written

as
r
=
Ar,
(10.28)
-
where
r
and
f
are
represented by the column matrices
r=
[
ii]
andF=
[
ii],
(10.29)
ORTHOGONAL TRANSFORMATIONS
169
and the transformation matrix
A
is
represented by the square matrix
(10.30)
We use both boldface letter
r
and
?;’

to denote
a
vector. Generalization
of
these formulas to
n
dimensions
is
again obvious. Transpose
of
a
matrix is
obtained by interchanging its rows and columns
as
(10.31)
-
r=
[
z1
zz
23
3
and
We can now write the magnitude
of
a
vector
as
-
rr=

[
z1
22 23
1
[
=z,q+.g+z;.
(10.32)
(10.33)
The magnitude
off
is
now given as
-
(10.34)
_&
rr
=F(xA>r,
where we have used the matrix property
D=BX.
(10.35)
From Equation
(10.34)
it is seen that linear transformations that preserve the
length
of
a
vector must satisfy the condition
XA
=
I,

(10.36)
where
I
is
the identity matrix
I=[:
s]
(10.37)
Such transformations are called orthogonal transformations. In terms of com-
ponents the orthogonality condition
[Eq.
(10.36)]
can be written
as
170
COORDINATES AND
TENSORS
Taking the determinant of the orthogonality relation, we see that the deter-
minant of transformations that preserve the length
of
a
vector satisfies
[DetAI2
=
1,
(10.39)
thus
DetA
=
fl.

(10.40)
Orthogonal transformations are basically transformations among Cartesian
coordinates without
a
scale change. Transformations with
DetA
=
1
are
called
proper
transformations.
They are composed of rotations and trans
lations. Transformations with
DetA
=
-1
are
called
improper transfor-
mations,
and they involve reflections.
10.2.1
Rotations About
Cartesian
Axes
For rotations about the xa-axis the rotation matrix takes the form
R3
=
(10.41)

01
Using the direction cosines we can write
&(0)
for counterclockwise rotations
as
(Fig.
10.5)
(10.42)
1
i0
01
0
COS~
sin4]andR2(+)=[ sin+
0
0
1
cos$
0
COSB
sin8
0
Rs(0)
=
-sin0 cos0
0
.
Similarly, the rotation matrices corresponding to counterclockwise rotations
about the
21-

and xa-axis can be written, respectively,
as
10
0
cos+
0
-sin+
0
-sin4
COS~
(10.43)
10.3
FORMAL PROPERTIES
OF
THE
ROTATION MATRIX
i)
Two
sequentially performed rotations, say
A
and
B,
is equivalent to
another rotation
C
as
C
=
AB.
(10.44)

ii) Because matrix multiplications
do
not commute, the order of rotations
is important, that
is,
in general
AB fBA.
(10.45)
FORMAL PROPER TIES
OF
THE
170
JA JlON MATRIX
171
-
3
NX1
fig.
10.5
Direction cosines
However, the associative law,
A(BC)
=
(AB)C,
(10.46)
holds between any three rotations
A,
B,
and
C.

nality relation
[Eq.
(lo.%)],
it is equal to the transpose of
A,
that is,
iii) The inverse transformation matrix
A-'
exists, and
from
the orthogo-
A-l=;i.
(10.47)
Thus
for
orthogonal transformations we can write
AA
=
AX
=
I.
(10.48)
172
COORD/NATES AND TENSORS
10.4
EULER ANGLES AND ARBITRARY ROTATIONS
The most general rotation matrix has nine components
(10.30).
However, the
orthogonality relation

AA
=
I,
written explicitly
as
-
(10.49)
gives
six relations among these components. Hence, only three
of
them can be
independent. In the study
of
rotating systems to describe the orientation
of
a
system it is important to define a set of three independent parameters. There
are a number
of
choices. The most common and useful are the three Euler
angles. They correspond to three successive rotations about the Cartesian
axes
so
that the final orientation
of
the system is obtained. The convention we
follow is the most widely used one in applied mechanics, in celestial mechanics,
and frequently, in molecular and solid-state physics. For different conventions,
we refer the reader to Goldstein
et

al.
The sequence starts with a counterclockwise rotation by
q5
about the x3-axis
of
the initial state
of
the system
as
This is followed by a counterclockwise rotation by
8
about the
xi
of
the
intermediate axis
as
Finally, the desired orientation is achieved by
a
counterclockwise rotation
about the x!-axis by
$I
as
EULER ANGLES AND ARBITRARY ROTATIONS
173
A(+),
B(6),
and
C(+)
are the rotation matrices for the corresponding trans-

formations, which
are
given
as
(10.53)
(10.54)
1
[o
01
"1
1
[o
01
cos+
sin+
0
B(+)=
-sin+
cosq5
0
,
0
-sin8 cos8
10
(10.55)
0
cos8
sin8
,
cos+

sin$
0
D(+)
=
-sin+
cos+
0
.
In terms of the individual rotations, elements of the complete transformation
matrix can
be
written
as
A
=
DCB,
(10.56)
A=
(10.57)
1
cos+cos
+
-
cos8
sin +sin
+
-
sin
+
cos

+
-
cos
6sin
+cos
+
cos+ sin
+
+
cos8cos+sin
+
-
sin
+
sin
+
+
cos
6cos
+cos
+
sin+sin6
cos
+sin
8
sin
8
sin
+
-sin

8cos
q5
cos
8
A-l
=
x.
(10.58)
We can also consider the elements
of
the rotation matrix
as
a
function of
The inverse of
A
is
some single parameter
t
and write
q5
=
w+t,
w
=
wet,
$
=
w&.
If

t
is taken
as
time,
w
can be interpreted
as
the constant angular velocity
about the corresponding axis. Now, in general the rotation matrix can
be
written
as
I
all(t) al2(t) al3(t)
a3l(t) a32(t) a33(t)
a21(t) a22(t) a23(t)
.
(10.59)
Using trigonometric identities it can
be
shown that
A(t2
+ti)
=
A(t2)A(ti).
(10.60)
Differentiating with respect to
t2
and putting
t2

=
0
and
tl
=
t,
we obtain
a
result that will be useful to
us
shortly
as
A'(t)
=
A'(O)A(t).
(10.61)
174
COORDINATES AND TENSORS
2
+
r
/'
-*
r
f
Fig.
10.6
Passive and active
views
of

the rotation matrix
10.5
ACTIVE AND PASSIVE INTERPRETATIONS
OF
ROTATIONS
It is possible to view the rotation matrix
A
in
F=Ar
(10.62)
as
an operator acting on
r
and rotating it in the opposite direction, while
keeping the coordinate axes fixed
(Fig.
This
is
called the
active
view.
The case where the coordinate
axes
are rotated
is
called the
passive
view.
In principle both the active and passive views lead to the same result.
However,

as
in quantum mechanics, sometimes the active view may offer some
advantages in studying the symmetries of
a
physical system.
In the case
of
the active view, we also need
to
know how an operator
A
transforms under coordinate transformations. Considering
a
transformation
represented by the matrix
B,
we multiply both
sides
of
Equation
(10.62)
by
B
to write
10.6)
.
BI;
=
BAr.
(10.63)

Using
BB-I
=
B-~B
=
I,
(10.64)
we now write Equation
(10.63)
as
BF
=
BAB-lBr,
-
r'
=
A'r'.
In the new coordinate system
T
and
r
are related
by
(10.65)
(10.66)
(10.67)
-
r'
=
BF

INFINITESIMAL TRANSFORMATIONS
175
and
rl
=
Br.
(10.68)
Thus the operator
A'
becomes
A'
=
BAB-l.
(10.69)
This is called
similarity
transformation.
If B
is an orthogonal transfor-
mation,
we
then write
A'
=
BAB.
(10.70)
In terms
of
components this can also be written
as

a!
v
'
=
b2kbljUbl.
(10.71)
10.6
INFINITESIMAL TRANSFORMATIONS
A
proper orthogonal transformation depending on a single continuous param-
eter
t
can be shown
as
r(t)
=
A(t)r(O).
(10.72)
Differentiating and using Equation
(10.61)
we
obtain
-=
dr(t)
A'(t)r(O)
dt
=
A'(O)A(t)r(O)
=
Xr(t),

(10.73)
(
10.74)
(
10.75)
where
X
=
A'(0). (10.76)
Differentiating Equation (10.75) we can now obtain the higher-order deriva-
tives as
(10.77)
Using these in the Taylor series expansion
of
r(t)
about
t
=
0
we write
(10.78)
176
COORDINATES AND TENSORS
thus obtaining
(10.79)
1
r(t)
=
(I
+

Xt+zX2t2
+
.
. .
)r(O)
.
This series converges, yielding
r(t)
=
exp(Xt)r(O)
.
(10.80)
This is called the exponential form of the transformation matrix. For infinites
imal transformations
t
is small; hence we can write
r(t)
N
(I
+
Xt)r(O),
(10.81)
(10.82)
Sr
~li
Xtr(O),
(10.83)
where
X
is called the generator

of
the infinitesimal transformation.
Using the definition
of
X
in Equation (10.76) and the rotation matrices
[Eqs. (10.42) and (10.43)] we obtain the generators of infinitesimal rotations
about the
z1- ,z2-
,
and
z~-axes,
respectively as
r(t)
-
r(0)
N
Xtr(O),
000
0 0
-1
0
10
(10.84)
An arbitrary infinitesimal rotation by the amounts
tl, t2,
and
t3
about their
respective axes can be written

as
r
=
(I
+
X3t3)(I
+
X2t2)(I
+
Xltl)r(O)
=
(I
+
X3t3
+
X2t2
+
X,t,)r(O).
(10.85)
Defining the vector
x
=
XIS,
+
X2S2
+
X3S3
=
(Xl,
x2,

X3)
(10.86)
and the unit vector
h
1
n=
dwi
[
i,]
(10.87)
we can write Equation (10.85)
as
r(t)
=
(I
+
X.iit)r(O),
(10.88)
where
t=
Jm.
(10.89)
This is an infinitesimal rotation about an axis in the direction
i?
by the amount
t.
For finite rotations we write
r(t)
=
ex-%(0).

(10.90)
INFINITESIMAL TRANSFORMATIONS
177
10.6.1
Infinitesimal Transformations Commute
Two successive infinitesimal transformations by the amounts tiand
t2
can be
written as
r
=
(I
+
X2t2)(I+ Xltl)r(O)
,
=
[I
+
(tlXl+t2X2)] r(0)
.
(10.91)
Because matrices commute with respect to addition and subtraction, infinites-
imal transformations also commute, that
is
For
finite rotations this is clearly not true. Using Equation (10.43) we can
write the rotation matrix for a rotation about the xz-axis followed by
a
rota-
tion about the xl-axis

as
1
0
-
sin
$
cosCp sin4cos$
cosdsin+ -sin4 cos~cos$
(10.93)
Reversing the order we get
cos
$
(10.94)
sin
$
sin
Cp
-
sin
$
cos
Cp
R2R1
=
0
cos
Cp
sin
4
[

sin
$
-
cos
$J
sin
q5
cos
$
cos
4
It
is
clear
that for finite rotations these two matrices are not equal:
RiR2
#
R2R1.
(10.95)
However, for small rotations, say by the amounts
Sic,
and
64,
we can use the
approximations
sin
6$
N
S+,
sin

6Cp
N
6Cp
(10.96)
cos6$
N
1, cos6Cp
_N
1
to find
10
R1R2
=
[
6+
1
=R2R,.
(10.97)
6$
-64
Note that in terms of the generators
[Eq.
(10.84)] we can also write this
as
100
0
0
-1 000
010
+6$00

0
+64
0
0
R1R2
=[o
0
0
0]
[o
-1
A]
=
I
+
S$Xl+
64x2
=
I
+
64x2
+
6+Xl
=
R2R1,
(10.98)